document.write( "Question 1188706: Prove by induction and through divisibility algorithm that 11^n - 6 is divisible by 5 for every positive integer n. \n" ); document.write( "
Algebra.Com's Answer #819855 by ikleyn(52790)\"\" \"About 
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\n" ); document.write( "Prove by induction and through divisibility algorithm that 11^n - 6 is divisible by 5 for every positive integer n.
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document.write( "(1)  Base case n= 1.\r\n" );
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document.write( "     Then  \"11%5En-6\" = 11 - 6 = 5  is divisible by 5,  so the base of induction is established.\r\n" );
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document.write( "(2)  The induction step from n to (n+1).\r\n" );
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document.write( "     We assume that  \"11%5En-6\"  is divisible by 6 for some integer positive index n.\r\n" );
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document.write( "     Then  \"11%5E%28n%2B1%29-6\" = \"11%2A11%5En+-+6+\" = \"11%2A%2811%5En-6%29+%2B+11%2A6+-+6\" = \"%2811%2A%2811%5En-6%29%29\" + (66-6) = \"11%2A%2811%5En-6%29\" + 60.\r\n" );
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document.write( "     The addend  \"11%2A%2811%5En-6%29\"  is divisible by 5 due to the inductive assumption, and the term 60 is also divisible by 5.\r\n" );
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document.write( "     Thus the inductive step from n to (n+1) is complete.\r\n" );
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document.write( "(3)  Due to the principle of Mathematical induction, the statement is proved for all positive integer n.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Above was the proof by induction.\r
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\n" ); document.write( "\n" ); document.write( "Below is more simple proof using the divisibility by 5 rule.\r
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document.write( "The number  \"11%5En\"  has the last (the units) digit  1 (one).\r\n" );
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document.write( "When we subtract 6 from this number, we get the last digit 5 for the difference,\r\n" );
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document.write( "which means that this difference,  \"11%5En-6\",  is divisible by 5 without a remainder.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved  (twice,  by two different methods).\r
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